Quantum interference in transport through almost symmetric double quantum dots
QQuantum interference in transport through almost symmetric double quantum dots
Zeng-Zhao Li and Martin Leijnse Division of Solid State Physics and NanoLund,Lund University, Box 118, S-22100 Lund, Sweden (Dated: March 4, 2019)We theoretically investigate transport signatures of quantum interference in highly symmetric dou-ble quantum dots in a parallel geometry and demonstrate that extremely weak symmetry-breakingeffects can have a dramatic influence on the current. Our calculations are based on a master equa-tion where quantum interference enters as non-diagonal elements of the density matrix of the doublequantum dots. We also show that many results have a physically intuitive meaning when recastingour equations as Bloch-like equations for a pseudo spin associated with the dot occupation. In theperfectly symmetric configuration with equal tunnel couplings and orbital energies of both dots,there is no unique stationary state density matrix. Interestingly, however, adding arbitrarily smallsymmetry-breaking terms to the tunnel couplings or orbital energies stabilizes a stationary stateeither with or without quantum interference, depending on the competition between these two per-turbations. The different solutions can correspond to very different current levels. Therefore, ifthe orbital energies and/or tunnel couplings are controlled by, e.g., electrostatic gating, the doublequantum dot can act as an exceptionally sensitive electric switch.
I. INTRODUCTION
Quantum interference is not only conceptually inter-esting, but also has a wide range of applications inquantum science and technology. Superconducting quan-tum interference devices are already standard technol-ogy, but quantum interference in electronic transport innanostructures might be beneficial in as diverse applica-tions as transistors and heat engines. Molecular junc-tions and artificial double-quantum-dot molecules havebeen widely investigated experimentally as typical plat-forms for demonstrating quantum interference of elec-trons.
Theoretical works have proposed that quantuminterference, in contrast to dynamical blockade inducedby electron-electron correlations, is responsible fora bunching behavior (super-Poisson noise or Fano fac-tor larger than 1) of electrons in transport, whichmight be useful for quantum communication with en-tangled electrons. Triple quantum dots or a singledot with three orbitals allows using quantum interfer-ence for demonstrating coherent population trapping ofelectrons.
The formed trapping state – or dark state– is potentially useful for cooling a nanomechanical res-onator. In devices where multiple quantum dots or moleculesare parallel-coupled to two leads, the importance of quan-tum interference depends in a complicated way on therelative strengths and phases of the various different tun-nel couplings. The dependence on the energy differencebetween the dot orbitals seems less complicated, withquantum interference being suppressed whenever this en-ergy difference exceeds the energy scale set by the tunnelcoupling. The quasi-degenerate case of orbital splittingand tunnel coupling of comparable magnitude was in-vestigated in Ref. 28 by deriving a master equation inthe singular-coupling limit. Note, however, that a laterwork, that focused on interference effects in a benzenemolecule, showed that the master equation in the pres- ence of quasidegenerate states could also be derived un-der the widely used weak-coupling approximation. Here, we focus on the case of an almost fully symmet-ric system, including only small perturbations away fromorbital degeneracy and tunnel-coupling symmetry. Wecalculate the stationary state electric current through aninteracting double quantum dot parallel-coupled betweentwo leads using a master equation approach. The masterequation is solved for the (reduced) density matrix de-scribing the nonequilibrium state of the double dot. Tomake a clear connection to previous works on transportin quantum dot spin-valves (see, e.g., Refs. 31 and 32),we recast the master equation as a Bloch-like equationfor a pseudo-spin representation of the dot occupation.In the fully symmetric case (equal energies of the twoquantum dot orbitals and equal tunnel couplings to bothdots and both leads), the rate matrix appearing in themaster equation becomes singular, signaling that thereis no unique stationary state (a similar instability in adifferent model was studied in Ref. 33). This case is,however, not experimentally relevant as energies and cou-plings will never be exactly equal. We therefore investi-gate the effects of weak perturbations to these param-eters, where weak means much smaller than all otherenergy scales of the system. We find that the interplaybetween perturbations to orbital degeneracy and tunnelcouplings dramatically affects the signatures of quantuminterference in electron transport. In the presence ofa small breaking of orbital degeneracy, but fully sym-metric tunnel couplings, there is no quantum interfer-ence between electrons tunneling through the two dots(the density matrix of the double quantum dot, writ-ten in the basis of occupation of the individual dots,is diagonal). The current is independent of the mag-nitude of the breaking of orbital degeneracy, as long asit remains the smallest energy scale of the problem. Incontrast, when the symmetry of the tunnel couplings isweakly broken, but the orbital degeneracy remains ex- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r act, the current is significantly suppressed due to de-structive interference (non-diagonal elements of the den-sity matrix). Again, the current is independent of thestrength and exact details of the symmetry breaking (aslong as it remains weak). When both the orbital de-generacy and the tunnel-coupling symmetry are weaklybroken, the current becomes highly sensitive to the de-tails and strengths of the different symmetry-breakingterms. In particular, for certain ranges of orbital ener-gies, extremely small variations in the tunnel couplingsor orbital energies switch the current between very closeto zero and a large value by turning quantum interfer-ence on or off. These changes can be induced by gatevoltages and such sensitive switching behavior – not lim-ited by temperature – is highly desirable in transistorsand related applications. We also find strong negativedifferential resistance and rectifying behavior.This paper is organized as follows. In Sec. II, we intro-duce our double quantum dot model. In Sec. III, a mas-ter equation approach and also the formalism of a pseudospin are presented. In Sec. IV, we demonstrate that thereis no unique stationary state solution for the completelysymmetric case and explain how the singularity appearsin the pseudo-spin formalism. The symmetry-breakingeffects due to small orbital detuning and tunnelling devia-tion on the quantum interference are illustrated in Sec. V.Summary and conclusions are finally given in Sec. VI. II. MODEL
The system of two single-orbital quantum dots in aparallel geometry, coupled to both left and right leadsis schematically shown in Fig. 1, and modelled by theHamiltonian (we use (cid:126) = k B = 1 throughout the rest ofthe paper) H = H S + H B + H T , (1)where H S = (cid:88) i =1 , ε i d † i d i + U d † d d † d , (2) H B = (cid:88) k,s =L , R ω k,s c † k,s c k,s , (3) H T = (cid:88) i =1 , (cid:88) k,s =L , R t k,s,i c † k,s d i + t ∗ k,s,i d † i c k,s . (4)Here, H S with fermionic operators d † i and d i describes thetwo single-orbital quantum dots with ε i being the onsiteorbital energy of the i th dot and U the Coulomb interac-tion between two electrons occupying different quantumdots. For conceptual simplicity we neglect the spin de-gree of freedom, knowing that the density matrix in thespinful case will remain diagonal in spin for nonmagneticleads and in the absence of spin-orbit coupling. The leftand right leads are described by H B with fermionic oper-ators c † k,s and c k,s . In the tunnel Hamiltonian H T , t k,s,i FIG. 1: (Color online) Two single-orbital quantum dotscoupled in parallel between two leads. The symmetry ofthe system can be broken by lifting the orbital degeneracy( δε = ε − ε (cid:54) = 0) and/or by unequal tunnelling rates ( δ j (cid:54) = 0, j = 0 , , , is the tunnelling amplitude between the i th quantum dotand the s th lead. We emphasize that the double quantumdot considered in our work is a parallel coupled doubledot (rather than a serial coupled) and that there is notunnel coupling between the dots.The eigenstates of the isolated double quantum dot are {| a (cid:105) , | b (cid:105) , | c (cid:105) , | d (cid:105)} with | a (cid:105) = | (cid:105) , | b (cid:105) = | (cid:105) , | c (cid:105) = | (cid:105) ,and | d (cid:105) = | (cid:105) where n and n in | n n (cid:105) represent theoccupation numbers of the 1st and 2nd dots, respectively.The double dot Hamiltonian in this basis becomes H S = ε | b (cid:105)(cid:104) b | + ε | c (cid:105)(cid:104) c | + ( ε + ε + U ) | d (cid:105)(cid:104) d | , (5)while the tunnel Hamiltonian is H T = (cid:88) k,s =L , R [ t ∗ k,s, ( | b (cid:105)(cid:104) a | + | d (cid:105)(cid:104) c | )+ t ∗ k,s, ( | c (cid:105)(cid:104) a | − | d (cid:105)(cid:104) b | )] c k,s + H.c.. (6)
III. MASTER EQUATION
Following the standard procedures such as the second-order perturbation theory, the Born-Markovian approx-imation, and tracing out the degree of freedom of theleads , the evolution of reduced double dot densitymatrix is governed by the equation˙ ρ = − i [ H S , ρ ] + L (cid:3) ρ + L (cid:2) ρ + P ρ, (7)where H S is given in Eq. (5) and the superoperators de-scribing dissipation of the system due to the couplings tothe leads are L (cid:3) ρ = (cid:88) s =L , R Γ s { f s ( ε ) D [ | b (cid:105)(cid:104) a | ] ρ + ¯ f s ( ε ) D [ || a (cid:105)(cid:104) b | ] ρ + f s ( ε + U ) D [ | d (cid:105)(cid:104) c | ] ρ + ¯ f s ( ε + U ) D [ | c (cid:105)(cid:104) d | ] ρ } + (cid:88) s =L , R Γ s { f s ( ε ) D [ | c (cid:105)(cid:104) a | ] ρ + ¯ f s ( ε ) D [ || a (cid:105)(cid:104) c | ] ρ + f s ( ε + U ) D [ | d (cid:105)(cid:104) b | ] ρ + ¯ f s ( ε + U ) D [ | b (cid:105)(cid:104) d | ] ρ } , (8) L (cid:2) ρ = (cid:88) s =L , R (cid:112) Γ s Γ s { f s ( ε ) D [ | c (cid:105)(cid:104) a | , | b (cid:105)(cid:104) a | ] ρ + ¯ f s ( ε ) D [ | a (cid:105)(cid:104) c | , | a (cid:105)(cid:104) b | ] ρ − f s ( ε + U ) D [ | d (cid:105)(cid:104) b | , | d (cid:105)(cid:104) c | ] ρ − ¯ f s ( ε + U ) D [ | b (cid:105)(cid:104) d | , | c (cid:105)(cid:104) d | ] ρ } + (cid:88) s =L , R (cid:112) Γ s Γ s { f s ( ε ) D [ | b (cid:105)(cid:104) a | , | c (cid:105)(cid:104) a | ] ρ + ¯ f s ( ε ) D [ | a (cid:105)(cid:104) b | , | a (cid:105)(cid:104) c | ] ρ − f s ( ε + U ) D [ | d (cid:105)(cid:104) c | , | d (cid:105)(cid:104) b | ] ρ − ¯ f s ( ε + U ) D [ | c (cid:105)(cid:104) d | , | b (cid:105)(cid:104) d | ] ρ } , (9)and P ρ = iπ (cid:88) s =L , R Γ s { p s ( ε )[ | a (cid:105)(cid:104) a | − | b (cid:105)(cid:104) b | , ρ ]+ p s ( ε + U )[ | c (cid:105)(cid:104) c | − | d (cid:105)(cid:104) d | , ρ ] } +Γ s { p s ( ε )[ | a (cid:105)(cid:104) a | − | c (cid:105)(cid:104) c | , ρ ]+ p s ( ε + U )[ | b (cid:105)(cid:104) b | − | d (cid:105)(cid:104) d | , ρ ] } + (cid:88) s =L , R (cid:112) Γ s Γ s ×{ p s ( ε )([ | a (cid:105)(cid:104) c | , [ | b (cid:105)(cid:104) a | , ρ ]] − [ | c (cid:105)(cid:104) a | , [ | a (cid:105)(cid:104) b | , ρ ]]) − p s ( ε + U )([ | b (cid:105)(cid:104) d | , [ | d (cid:105)(cid:104) c | , ρ ]] − [ | d (cid:105)(cid:104) b | , [ | c (cid:105)(cid:104) d | , ρ ]])+ p s ( ε )([ | a (cid:105)(cid:104) b | , [ | c (cid:105)(cid:104) a | , ρ ]] − [ | b (cid:105)(cid:104) a | , [ | a (cid:105)(cid:104) c | , ρ ]]) − p s ( ε + U )([ | c (cid:105)(cid:104) d | , [ | d (cid:105)(cid:104) b | , ρ ]] − [ | d (cid:105)(cid:104) c | , [ | b (cid:105)(cid:104) d | , ρ ]]) } . (10)Explicit expressions for the different matrix elements of ρ are given in Appendix A. For the calculations pre-sented in this work, we have in part relied on the numer-ical implementation described in Ref. 36. In Eq. (10), p s ( ω ) = (cid:60) [Ψ( + i π ω − µ s T s )], where (cid:60) [ · ] denotes the realpart and the digamma function Ψ originates from princi-pal value integrals. µ s and T s are the chemical potentialand temperature of lead s . In all calculation we considera symmetric voltage bias, µ L = − µ R = eV b /
2, and equaltemperatures, T L = T R = T . f s ( ω ) in Eqs. (8) and (9) isthe Fermi-Dirac distribution and ¯ f s ( ω ) = 1 − f s ( ω ). Wehave furthermore defined D [ A ] ρ = 2 AρA † − ρA † A − A † Aρ, (11) D [ A, B ] ρ = AρB † + BρA † − ρB † A − A † Bρ. (12)Note that D [ A ] ρ = D [ A, A ] ρ which only involves a singlepathway of electron tunnelings. The bare tunnelling rateis Γ si = π | t k,s,i | (cid:37) s with (cid:37) L(R) being the density of statesin the left (right) lead which is assumed to be constant.We parametrize the tunnel couplings as (see Fig. 1)Γ L = Γ + δ , Γ R = Γ + δ , (13)Γ L = Γ + δ , Γ R = Γ + δ , (14)where δ j ( j = 0 , , ,
3) is a (small) perturbation to thetunnelling rate Γ. We also let δε = ε − ε denote the energy difference between dot orbitals and ε = ( ε + ε ) / ε = − eαV g , where we set α = 1 for simplicity.In the fully symmetric case we have δε = δ j = 0.In Eq. (7), the first term describes the free evolution ofthe double quantum dot. L (cid:3) ρ given by Eq. (8) involvesthe well-known form D [ A ] ρ [Eq. (11)] and correspondsto tunnelling processes without interference of electrons.Thus, including only this term is equivalent to the Paulirate equation for the diagonal stationary density matrix[the first term in Eq. (7) vanishes in this case]. In addi-tion, Eq. (7) explicitly contains L (cid:2) ρ [Eq. (9)] involving D [ A, B ] ρ ( A (cid:54) = B ) that is responsible for quantum coher-ence between states where an electron occupies dot 1 anddot 2, mediated by the electrodes. Note that L (cid:2) ρ wouldnot survive the rotating-wave approximation for large δε .Finally, Eq. (7) includes the term P ρ [Eq. (10)] whichoriginates from principal value integrals and describes atunneling-induced shift of the quantum dot orbitals.The current operator is given by I s = ie (cid:80) k,s,i t k,s,i c † k,s d i − H.c. . Here we consider thestationary state current flowing into lead R, I = (cid:104) I R (cid:105) which is given by I − e = 2 { [Γ R1 ¯ f R ( ε + U ) + Γ R2 ¯ f R ( ε + U )] ρ dd − [Γ R1 f R ( ε ) + Γ R2 f R ( ε )] ρ aa +[Γ R1 ¯ f R ( ε ) − Γ R2 f R ( ε + U )] ρ bb +[Γ R2 ¯ f R ( ε ) − Γ R1 f R ( ε + U )] ρ cc } + (cid:112) Γ R1 Γ R2 [ ¯ f R ( ε ) + ¯ f R ( ε )+ f R ( ε + U ) + f R ( ε + U )]( ρ bc + ρ cb ) , (15)where ρ αβ ( α, β = a, b, c, d ) are the elements of the sta-tionary state density matrix.In addition to the standard form of the master equa-tion, we consider Bloch-like equations for pseudo-spincomponents in order to help understand and interpret ourresults. We define the z component of the pseudo spinas being proportional to the charge difference betweenthe two dots. Similar to the case of a real spin, therelation between pseudo-spin components and reduceddensity matrix elements is then given by S x = ρ bc + ρ cb , S y = i ρ bc − ρ cb , S z = ρ bb − ρ cc . (16)The spin components also couple to the populations, P = ρ aa , P = ρ bb + ρ cc and P = ρ dd . Insteadof { ρ aa , ρ bb , ρ cc , (cid:60) [ ρ bc ] , (cid:61) [ ρ bc ] , ρ dd } , we now could have anew set of variables { P , P , S z , S x , S y , P } . Then themaster equation for density matrix elements is equiva-lent to the combination of rate equations for populations P , P , P and Bloch-like equations for the spin compo-nents, see Appendix C. IV. FULLY SYMMETRIC TWOCAPACITIVELY COUPLED QUANTUM DOTS
In this section we focus on the symmetric case of de-generate dot orbitals, δε = 0, and identical tunnellingcouplings, i.e., all δ j = 0 (see Fig. 1). In this case, thecommutator in Eq. (7) is zero and Eqs. (8) and (9) sim-plify to Eqs. (B1) and (B2) in Appendix B.We first present the result of solving the Pauli rateequations, which corresponds to including only the term L (cid:3) ρ in Eq. (7) and assuming the density matrix to bediagonal. The resulting stability diagram ( dI/dV b as afunction of V g and V b ) is shown in Fig. 2(a) and exhibitsthe typical Coulomb blockade behavior. At low V b , thenumber of electrons on the double quantum dot is fixedand no current flows, except close to the two charge de-generacy points at eV g = 0 and eV g = U . Increasing V g starting from negative values, the charge on the doublequantum dot is increased from zero to one and then totwo electrons. At larger V b , outside the Coulomb block-ade region, a current flows because of single-electron tun-neling.We now turn to a solution for the full, possibly non-diagonal density matrix. For the fully symmetric case,setting the time derivates to zero in the equations in Ap-pendices A and B yields a singular matrix equation. Thisshows that there is no unique stationary state. For thetime-dependent case, one would obtain a solution which,in principle, depends on the initial state regardless of howlong one waits.The singular behavior can be understood by noticingthat the equations for ρ bb and ρ cc are exactly the same,implying ρ bb = ρ cc . Because ρ cb = ρ ∗ bc must hold, thereare then only two independent equations for the threeindependent elements ρ bb , ρ bc , ρ dd (cid:88) s [ f s ( ε ) ρ aa − ¯ f s ( ε )( ρ bb + ρ bc )] , (17)0 = 4Γ (cid:88) s [ f s ( ε + U )( ρ bb − ρ bc ) − ¯ f s ( ε + U ) ρ dd ] , (18)where ρ aa = 1 − ρ bb − ρ dd . One can thus fix one densitymatrix element, say ρ bc , and then find a unique solutionfor the other elements. For example, the choice ρ bc = 0gives a solution without quantum interference which isthe same as would be obtained by solving the Pauli rateequation for the diagonal density matrix elements.Figure 2(b) shows the current obtained by solvingEqs. (17) and (18) with different choices for ρ bc andinserting the solution into Eq. (15). The curve with ρ bc = 0 corresponds to the current along a horisontal cutat eV b = 10 Γ in Fig. 2(a). Compared with this solution,quantum interference ( ρ bc (cid:54) = 0) can either enhance or re-duce the current. Note, however, that depending on thechoice of ρ bc the results can become clearly unphysical,with negative occupation probabilities and even currentsflowing against the applied bias voltage. We emphasizethat the unphysical results originate from choosing ρ bc FIG. 2: (Color online) Results for a completely symmetricdouble quantum dot. (a) Stability diagram ( dI/dV b as a func-tion of V g and V b ) obtained by solving the Pauli rate equationfor the diagonal elements of the density matrix. (b) Currentat eV b = 10 Γ as a function of V g (along a horisontal cut ina stability diagram like in (a)), obtained by solving Eqs. (17)and (18) for different choices of ρ bc . The parameters usedin both (a) and (b) are U = 2 × Γ, µ L = − µ R = eV b / T = 862Γ. when the steady-state equations do not have a uniquesolution. Importantly, the physical solutions we will ob-tain below in the presence of weak symmetry breakingeffects never display such unphysical behavior. This isbecause the corresponding equations given below havewell-determined steady-state solutions.The singular behavior can alternatively be understoodby reformulating the master equation in terms of thepseudo-spin components S x , S y , and S z [Eq. (16)], as wellas the populations P , P , and P . The explicit equationsare given in Appendix C. In the fully symmetric case, theequations for the spin components [Eqs. (C1)–(C3)] be-come˙ S x = − (cid:88) s [ ¯ f s ( ε ) + f s ( ε + U )] S x + Γ (cid:88) s { f s ( ε ) P − [ ¯ f s ( ε ) − f s ( ε + U )] P − f s ( ε + U ) P } , (19)˙ S y = − (cid:88) s { [ ¯ f s ( ε ) + f s ( ε + U )] S y + B sx S z } , (20)˙ S z = 2Γ (cid:88) s { B sx S y − [ ¯ f s ( ε ) + f s ( ε + U )] S z } . (21)Here, B sx = π [ p s ( ε ) + p s ( ε + U )] is an effective magneticfield in the x direction [see Eq. (10)]. Equations (C6)–(C8) for the populations become˙ P = 2Γ (cid:88) s [ − f s ( ε ) P + ¯ f s ( ε ) P + 2 ¯ f s ( ε ) S x ] , (22)˙ P = 2Γ (cid:88) s { f s ( ε ) P − [ ¯ f s ( ε ) + f s ( ε + U )] P +2 ¯ f s ( ε + U ) P −
2[ ¯ f s ( ε ) − f s ( ε + U )] S x } , (23)˙ P = 2Γ (cid:88) s [ f s ( ε + U ) P − f s ( ε + U ) P − f s ( ε + U ) S x ] . (24)Here, spin accumulation appears only in the x directionbecause only the spin component S x depends on the pop-ulations. In fact, there are two sets of equations whichdo not couple to each other, one set for { S x , P , P , P } and one for { S y , S z } . In the stationary state, Eqs. (20)and (21) imply that S y = S z = 0. Note that this alsomeans that there is no dependence on the principle valueintegrals.Similar to Eqs. (17) and (18) for the density matrix,after using P + P + P = 1, there are only two inde-pendent equations for the remaining three variables, forexample:˙ S x −
12 ˙ P = − (cid:88) s f s ( ε + U ) S x + Γ (cid:88) s { f s ( ε + U ) P − f s ( ε + U ) P } , (25)and ˙ S x + 12 ˙ P = − (cid:88) s ¯ f s ( ε ) S x + Γ (cid:88) s { f s ( ε ) P − f s ( ε ) P } . (26)Therefore, we can solve the equations if we choose thevalue of one parameter. For example, fixing S x wouldcorrespond to fixing ρ bc , and setting S x = 0 leads to thesame result as the Pauli rate equation.Further insight into the singular nature of the masterequation can be obtained by a change of basis for thestates with one electron on the double quantum dot | + (cid:105) = 1 √ | b (cid:105) + | c (cid:105) ) , (27) |−(cid:105) = 1 √ | b (cid:105) − | c (cid:105) ) . (28)These are analogous to the bonding and anti-bondingstates of an electron in two degenerate orbitals, but theyhere have the same energy because there is no direct tun-nel coupling between the orbitals. We now note that H T | + (cid:105) = (cid:88) k,s =L , R t k,s c † k,s √ | a (cid:105) , (29) H T |−(cid:105) = − (cid:88) k,s =L , R t ∗ k,s c k,s √ | d (cid:105) . (30)This shows that tunneling can only change the state ofthe double quantum dot either between | a (cid:105) (no electronson the double dot) and | + (cid:105) , or between | d (cid:105) (two elec-trons on the double dot) and |−(cid:105) . Thus, in this basisit becomes clear that the master equation is separatedinto two different sectors which are not connected toeach other through tunneling. Therefore, there can beno unique stationary state. V. SYMMETRY-BREAKING EFFECTS
We now consider small deviations away from perfectsymmetry, i.e., δε and/or δ j finite, but still much smallerthan all other energy scales of the problem. Based on our FIG. 3: (Color online) Current and spin components as afunction of V g obtained from different approximations to themaster equation. The parameter are δε = 0, δt = 10 − Γ, (cid:126)d = ( − , , , − U = 2 × Γ, eV b = 10 Γ, T = 862Γ. conclusions for the symmetric system, a number of ques-tions arise: Will arbitrarily small symmetry-breakingterms remove the singular behavior of the master equa-tion and restore a well-defined unique stationary state?Will this stationary state be the same regardless of thedetails of the perturbations, as long as they are small?Will the possible stationary state(s) be one (or a sub-set) of the possible states for the symmetric system, orsomething different?In the following, we first consider two limiting cases,breaking either the orbital degeneracy or the tunnellingcoupling symmetry. We then investigate the case whereboth symmetries are broken simultaneously. A. Breaking orbital degeneracy
We first consider symmetric tunnel couplings, δ j = 0,but a small breaking of the degeneracy of the quantumdot orbitals, δε (cid:28) T, U,
Γ. We find that the stationarymaster equation (or, equivalently, the Block-like equa-tion for the pseudo spin) becomes well defined for anarbitrarily small but finite δε . Moreover, the stationarydensity matrix is fully diagonal and the current is equiv-alent to that given by the Pauli rate equation for thediagonal density matrix. The solution is equal to thatin Fig. 2(a) and independent of δε as long as it remainsby far the smallest energy scale. It is well known thatthe off-diagonal elements of the density matrix vanishfor large orbital detuning ( δε (cid:29) Γ), but the result thatthey are zero even for an arbitrarily small detuning onlyholds when the tunnel couplings are fully symmetric.
B. Breaking tunnel coupling symmetry
We now turn to the case of complete orbital degen-eracy, δε = 0, but with some asymmetry in the tunnelcouplings which we parametrize as (see Fig. 1) δ i = δt × d i with d i of order 1 and δt (cid:28) T, U,
Γ. We will consider dif-ferent configurations of the tunnel coupling asymmetry,but note two special cases, (cid:126)d = (1 , , ,
1) (equivalent tothe symmetric case) and (cid:126)d = ( d , d , d , d ) (representingasymmetric left/right couplings, but equal couplings toboth quantum dots). Except these two special cases, wefind that an arbitrarily small tunnel coupling asymmetryleads to a well-defined master equation with a unique sta-tionary state. The explicit equations for the pseudo-spincomponents and populations for the case δε = 0, δt (cid:54) = 0are presented in Appendix C 1.Figure 3(a) shows the current as a function of V g [sim-ilar to Fig. 2(b)] for (cid:126)d = ( − , , , −
1) and δt = 10 − Γ,comparing the solution for the Pauli rate equation andthe master equation with and without including the prin-ciple value integrals. The stationary state is in this casenot described by a diagonal density matrix. Quantuminterference has a large impact on the current, which be-comes suppressed to almost zero over a large range in V g where the Pauli rate equation predicts a large current.Interestingly, we find that the stationary density matrixand current are independent of δt (as long as it remainsmuch smaller than all other energy scales) and indepen-dent of (cid:126)d (except for the two special cases mentionedabove). The results presented in Fig. 3 are therefore uni-versal for the case δε = 0 and δt (cid:28) T, U,
Γ.We can gain some understanding based on the states | + (cid:105) and |−(cid:105) defined in Eqs. (27) and (28). If we focusfirst on the V g range where current flow is associated withfluctuations between zero and one electron on the doubledot [left peak in Fig. 3(a)], Eqs. (29) and (30) show thatfor the completely symmetric case, transport can onlyinvolve fluctuations between states | a (cid:105) and | + (cid:105) . Thus, | + (cid:105) act as a bright state, while |−(cid:105) becomes effectivelydecoupled from the leads and acts as a dark state. Theoccupation of the states | + (cid:105) and |−(cid:105) are given by ρ ++ = 12 ( ρ bb + ρ cc + ρ bc + ρ cb ) = 12 P + S x , (31) ρ −− = 12 ( ρ bb + ρ cc − ρ bc − ρ cb ) = 12 P − S x . (32)When now introducing a small symmetry-breaking term,this gives rise to a small coupling between the |−(cid:105) stateand the leads. In Fig. 3(b) we see that this coupling re-sults in S x attaining large negative values over a range in V g that precisely correspond to the suppression of the leftcurrent peak (compared with the Pauli rate equation). Alarge negative S x corresponds to a large occupation of the |−(cid:105) state [Eq. (32)] which acts as a dark state in this V g range. Thus, the current is decreased because the darkstate, which is only very weakly coupled to the leads, be-comes occupied with large probability and blocks trans-port through the bright state because the large Coulomb energy U shifts the doubly occupied state | d (cid:105) high up inenergy.In the V g range corresponding to the right currentpeak, where current flow is associated with fluctua-tions between one and two electrons on the double dot,Eqs. (29) and (30) show that instead the | + (cid:105) state is thedark state, while |−(cid:105) is the bright state. Here, the sup-pression of the current is instead associated with a sig-nificant positive S x , meaning a large occupation of | + (cid:105) .We can also understand the large impact of the prin-ciple value integrals (which are often neglected in masterequations). Figure 3(b) shows that the principle value in-tegrals reduce | S x | over a range in V g , leading to a largercurrent compared to the case where they are neglected.The reason for the decrease in | S x | is the pseudo mag-netic field in Eqs. (C9)–(C11) which allows the spin torotate away from the x axis (or the electron to escapefrom the dark state).Figures 3(c) and (d) show S y and S z , which remainmuch smaller than S x for all V g . C. Breaking both orbital degeneracy and tunnelcoupling symmetry
We now turn to the general case with δε (cid:54) = 0 and δt (cid:54) = 0, but still δε, δt (cid:28) T, U,
Γ. Figure 4 shows thestability diagrams, current and density matrix elementsas a function of V g for three different choices of (cid:126)d (mean-ing different configurations of the asymmetry in tunnelcouplings) for δt = δε = 10 − Γ. The current clearlydeviates from that given by the Pauli rate equation and,unlike the case with δε = 0, depends on the specific choiceof (cid:126)d (the result of the Pauli rate equation is, in contrast,independent of (cid:126)d as long as δt (cid:28) T, U,
Γ). We also notethat including the principle value integrals has a largeimpact on the current, which can be either enhanced orsuppressed compared with the (commonly used) masterequation where these terms are neglected.Interestingly, the stability diagrams in Figs. 4(a), (d)and (g) all show regions of negative differential resistance,as well as strong rectifying behavior (different magnitudeof the current for positive and negative V b ). Such effectsare expected in quantum dot systems when the tunnelcouplings to different orbitals and/or leads differ sub-stantially, but here they are induced by quantum inter-ference even in the case where the system is very close tosymmetric.Figure 5 shows the current as a function of V g for fixed δt and (cid:126)d , but with increasing δε . For δε (cid:28) δt the cur-rent is suppressed over a large range in V g , similar toFig. 2. The current starts to recover when δε ∼ δt andapproaches the value given by the Pauli rate equationfor δε (cid:29) δt (even though δε (cid:28) Γ still holds). As longas δε, δt (cid:28)
T, U,
Γ holds, the current depends only onthe ratio δε/δt (and on (cid:126)d if δε ∼ δt ). The dependenceon δε/δt is shown in Fig. 6 where we plot the current as FIG. 4: (Color online) Stability diagrams [(a), (d) and (g)], current as a function of V g with eV b = 10 Γ [(b), (e) and (h)] anddensity matrix elements as a function of V g with eV b = 10 Γ [(c), (f) and (i)]. (cid:126)d is varied while keeping the length of the vectorfixed, (cid:126)d = ( − , , , −
1) in (a)–(c), (cid:126)d = (0 , √ , ,
1) in (d)–(f) and (cid:126)d = (0 , , ,
2) in (g)–(i). Here δt = δε = 10 − Γ and all otherparameters are the same as in Fig. 3. a function of δε and δt for fixed V g = 0. The currentswitches between its two limiting values along a diagonalline, which shows that very small controlled changes in δε or δt can have a large impact on the current. VI. SUMMARY AND CONCLUSIONS
In this work, we have shown that for a double quantumdot with degenerate orbitals and equal tunnel couplingsto both dots and both leads, the master equation for thestationary density matrix becomes singular. This sin-gular behavior implies that a whole family of stationarystates is allowed by the equations, in principle, one of them being the diagonal density matrix as obtained fromthe Pauli rate equation. When including perturbationsaway from the perfectly symmetric system, the stationarystate becomes unique. When considering only breakingof orbital degeneracy (but symmetric tunnel couplings),this stationary state is a diagonal density matrix, evenif the orbital detuning is much smaller than all otherenergy scales. In the opposite case of orbital degener-acy but small perturbations to the tunnel couplings, thedensity matrix is non-diagonal and the current is signif-icantly suppressed by quantum interference. In the caseof breaking both orbital degeneracy and tunnel couplingsymmetry, the current is a very sensitive function of theratio of the symmetry breaking terms.
FIG. 5: (Color online) Current as a function of V g with eV b =10 Γ for different values of δε at fixed δt = 10 − Γ and (cid:126)d =( − , , , − δε and δt atfixed eV g = 0 with eV b = 10 Γ. All other parameters are thesame as in Fig. 3.
We note that our study assumes that both quantumdots are coupled to the same lead states. This is a goodassumption roughly when the points where the two dotscouple to a given lead are separeted by less than the Fermiwave length in the leads. This condition can be realizedwith semiconductor leads (large Fermi wavelength) or byusing a molecular double quantum dot (small distancebetween dots). The effects of increasing the distance be-tween the quantum dots were investigated in 37,38, whereit was shown to destroy quantum interference features.We have relied on a master equation which is basedon leading order perturbation theory in the tunnel cou- plings, which is expected to be a very good approxima-tion in all cases we consider because we stay in the regimeΓ (cid:28) T . Nonetheless, we have used the methods de-scribed in Refs. 36,39 to verify that processes which arenext-to-leading order in the tunneling coupling do notaffect our conclusions. It would be interesting to, in a fu-ture study, investigate the regime Γ (cid:46) T where cotunnel-ing and other higher order tunnel processes become im-portant. Note that our Hamiltonian of two capacitivelycoupled quantum dots in terms of spinless fermionic op-erators resembles a Hubbard site for spinful electrons. Itis therefore possible to reformulate the model consideredin the present work in terms of a single dot with spinup and spin down electrons coupled to partially spin-polarized leads, as implied by a pseudo spin defined inEq. (16). For strong tunnel couplings, one would ex-pect the appearance of Kondo effect associated with thepseudo spin. In the present work we focus instead on theregime of weak tunnel coupling. It might also be interest-ing to consider shot noise or full counting statistics thatcould provide more information about electron transportin our system. Decoherence effects due to charge noise or phonons,not included in this study, would lead to a decay of elec-trons trapped in the dark state. However, the qual-itative results are not affected by weak decoherence.We note that a recent demonstration of Landau-Zener-St¨uckelberg-Majorana interferometry in a silicon-basedsingle-electron double quantum dot suggests that deco-herence can be weak enough for quantum interference tosurvive .From an application perspective, the double quantumdot system we investigate could be used as an extremelysensitive electric switch, similar to a field effect transis-tor. If an additional electrostatic gate is used to controlthe tunnel coupling asymmetry ( δt ) or the orbital detun-ing ( δε ), one can switch the current between almost zeroand a large finite value by a tiny shift in either of theseparameters which makes the system cross the diagonalin Fig. 6. Note that the change happens when chang-ing a parameter by an amount that is much smaller thanall other energy scales. In particular, unlike most otherswitches, this switch will not be limited by temperature. Acknowledgments
We acknowledge helpful discussions with Martin Josef-sson and Andreas Wacker, and financial support from theCarl Trygger Foundation and from NanoLund.
Appendix A: Explicit form of the Master equation
Starting from Eq. (7), the explicit forms of Eqs. (8)–(10) for the different density matrix elements are˙ ρ aa = (cid:88) s {− s f s ( ε ) + Γ s f s ( ε )] ρ aa + 2Γ s ¯ f s ( ε ) ρ bb +2Γ s ¯ f s ( ε ) ρ cc } + (cid:88) s (cid:112) Γ s Γ s { [ ¯ f s ( ε ) + ¯ f s ( ε )] × ( ρ cb + ρ bc ) + iπ [ p s ( ε ) − p s ( ε )]( ρ bc − ρ cb ) } , (A1)˙ ρ bb = (cid:88) s { s f s ( ε ) ρ aa − s ¯ f s ( ε ) + Γ s f s ( ε + U )] × ρ bb + 2Γ s ¯ f s ( ε + U ) ρ dd } − (cid:88) s (cid:112) Γ s Γ s { [ ¯ f s ( ε ) − f s ( ε + U )]( ρ cb + ρ bc ) − iπ [ p s ( ε ) + p s ( ε + U )] × ( ρ bc − ρ cb )] } , (A2)˙ ρ cc = (cid:88) s { s f s ( ε ) ρ aa − s ¯ f s ( ε ) + Γ s f s ( ε + U )] × ρ cc + 2Γ s ¯ f s ( ε + U ) ρ dd } − (cid:88) s (cid:112) Γ s Γ s { [ ¯ f s ( ε ) − f s ( ε + U )]( ρ bc + ρ cb ) + iπ [ p s ( ε + U ) + p s ( ε )] × ( ρ bc − ρ cb ) } , (A3)˙ ρ dd = (cid:88) s { s f s ( ε + U ) ρ bb + 2Γ s f s ( ε + U ) ρ cc − s ¯ f s ( ε + U ) + Γ s ¯ f s ( ε + U )] ρ dd }− (cid:88) s (cid:112) Γ s Γ s { [ f s ( ε + U ) + f s ( ε + U )]( ρ bc + ρ cb ) + iπ [ p s ( ε + U ) − p s ( ε + U )]( ρ bc − ρ cb ) } , (A4)˙ ρ bc = i ( ε − ε ) ρ bc + (cid:88) s (cid:112) Γ s Γ s { [ f s ( ε ) + f s ( ε )] ρ aa − [ ¯ f s ( ε ) − f s ( ε + U )] ρ bb − [ ¯ f s ( ε ) − f s ( ε + U )] × ρ cc − [ ¯ f s ( ε + U ) + ¯ f s ( ε + U )] ρ dd } − (cid:88) s { Γ s × [ ¯ f s ( ε ) + f s ( ε + U )] + Γ s [ ¯ f s ( ε ) + f s ( ε + U )] }× ρ bc + (cid:88) s iπ { Γ s [ p s ( ε ) + p s ( ε + U )] − Γ s × [ p s ( ε ) + p s ( ε + U )] } ρ bc + (cid:88) s (cid:112) Γ s Γ s iπ ×{ [ p s ( ε ) − p s ( ε )] ρ aa + [ p s ( ε ) + p s ( ε + U )] × ρ bb − [ p s ( ε + U ) + p s ( ε )] ρ cc + [ p s ( ε + U ) − p s ( ε + U )] ρ dd } . (A5) From these expressions one can verify that (cid:80) α = a,b,c,d ˙ ρ αα = 0, which is due to probabilitynormalization, (cid:80) α = a,b,c,d ρ αα = 1. Appendix B: Master equation for the fullysymmetric system
In the fully symmetric case with δε = 0 and all δ j = 0,Eqs. (8)–(10) become L (cid:3) ρ = (cid:88) s =L , R Γ s { f s ( ε )( D [ | b (cid:105)(cid:104) a | ] ρ + D [ | c (cid:105)(cid:104) a | ] ρ )+ ¯ f s ( ε )( D [ | a (cid:105)(cid:104) b | ] ρ + D [ | a (cid:105)(cid:104) c | ] ρ )+ f s ( ε + U )( D [ | d (cid:105)(cid:104) c | ] ρ + D [ | d (cid:105)(cid:104) b | ] ρ )+ ¯ f s ( ε + U )( D [ | c (cid:105)(cid:104) d | ] ρ + D [ | b (cid:105)(cid:104) d | ] ρ ) } , (B1) L (cid:2) ρ = (cid:88) s =L , R Γ s { f s ( ε ) D [ | b (cid:105)(cid:104) a | , | c (cid:105)(cid:104) a | ] ρ + ¯ f s ( ε ) D [ | a (cid:105)(cid:104) b | , | a (cid:105)(cid:104) c | ] ρ + f s ( ε + U ) D [ | d (cid:105)(cid:104) c | , | d (cid:105)(cid:104) b | ] ρ + ¯ f s ( ε + U ) D [ | c (cid:105)(cid:104) d | , | b (cid:105)(cid:104) d | ] ρ } , (B2) P ρ = iπ (cid:88) s =L , R Γ s { p s ( ε )([ | a (cid:105)(cid:104) a | − | b (cid:105)(cid:104) b | , ρ ]+[ | a (cid:105)(cid:104) a | − | c (cid:105)(cid:104) c | , ρ ] + [ | a (cid:105)(cid:104) c | , [ | b (cid:105)(cid:104) a | , ρ ]] − [ | c (cid:105)(cid:104) a | , [ | a (cid:105)(cid:104) b | , ρ ]] + [ | a (cid:105)(cid:104) b | , [ | c (cid:105)(cid:104) a | , ρ ]] − [ | b (cid:105)(cid:104) a | , [ | a (cid:105)(cid:104) c | , ρ ]]) − p s ( ε + U ) × ([ | c (cid:105)(cid:104) c | − | d (cid:105)(cid:104) d | , ρ ] + [ | b (cid:105)(cid:104) b | − | d (cid:105)(cid:104) d | , ρ ]+[ | b (cid:105)(cid:104) d | , [ | d (cid:105)(cid:104) c | , ρ ]] − [ | d (cid:105)(cid:104) b | , [ | c (cid:105)(cid:104) d | , ρ ]]+[ | c (cid:105)(cid:104) d | , [ | d (cid:105)(cid:104) b | , ρ ]] − [ | d (cid:105)(cid:104) c | , [ | b (cid:105)(cid:104) d | , ρ ]]) } . (B3)Here D [ A, B ] ρ is the sum of two different tunneling path-ways, which are equivalent due to the exact orbital de-generacy D [ A, B ] ρ = D [ A, B ] ρ + D [ B, A ] ρ = 2 AρB † + 2 BρA † − ρB † A − B † Aρ − ρA † B − A † Bρ. (B4)0
Appendix C: Bloch-like equations for the pseudospin
The Bloch-like equations for S x , S y , S z are˙ S x = − (cid:88) s { Γ s [ ¯ f s ( ε ) + f s ( ε + U )] + Γ s [ ¯ f s ( ε )+ f s ( ε + U )] } S x + [( ε − ε ) + (cid:88) s (Γ s B sz − Γ s B sz )] S y − (cid:88) s (cid:112) Γ s Γ s [ ¯ f s ( ε ) − f s ( ε + U )+ f s ( ε + U ) − ¯ f s ( ε )] S z + (cid:88) s (cid:112) Γ s Γ s { [ f s ( ε )+ f s ( ε )] P − [ ¯ f s ( ε + U ) + ¯ f s ( ε + U )] P − [ ¯ f s ( ε ) − f s ( ε + U ) − f s ( ε + U ) + ¯ f s ( ε )] P } , (C1)˙ S y = [( ε − ε ) + (cid:88) s (Γ s B sz − Γ s B sz )] S x − (cid:88) s { Γ s × [ ¯ f s ( ε ) + f s ( ε + U )] + Γ s [ ¯ f s ( ε ) + f s ( ε + U )] }× S y − (cid:88) s (cid:112) Γ s Γ s ( B sx + B sx ) S z − (cid:88) s (cid:112) Γ s Γ s × π { [ p s ( ε ) − p s ( ε )] P + [ p s ( ε + U ) − p s ( ε + U )] P +[ p s ( ε ) − p s ( ε + U ) + p s ( ε + U ) − p s ( ε )] P } , (C2)˙ S z = (cid:88) s (cid:112) Γ s Γ s { ( B sy − B sy ) S x + ( B sx + B sx ) S y }− (cid:88) s [Γ s ¯ f s ( ε ) + Γ s f s ( ε + U ) + Γ s ¯ f s ( ε )+Γ s f s ( ε + U )] S z + (cid:88) s { [Γ s f s ( ε ) − Γ s f s ( ε )] × P + [Γ s ¯ f s ( ε + U ) − Γ s ¯ f s ( ε + U )] P } + (cid:88) s { Γ s [ ¯ f s ( ε ) − f s ( ε + U )] − Γ s [ ¯ f s ( ε ) − f s ( ε + U )] } P . (C3)Here we defined the pseudo magnetic fields B sxi = B szi = 1 π [ p s ( ε i ) + p s ( ε i + U )] , (C4) B syi = ¯ f s ( ε i ) + f s ( ε i + U ) , (C5)where p s ( ω ) = (cid:60) [Ψ( + i π ω − µ s T s )] (see main paper). Notethat the pseudo magnetic field in the y direction van-ishes in Eqs. (C1)–(C2) when δε = 0, while the pseudomagnetic field in the z direction vanishes only when δt = δε = 0. The rate equations for the populations read˙ P = (cid:88) s {− s f s ( ε ) + Γ s f s ( ε )] P + [Γ s ¯ f s ( ε )+Γ s ¯ f s ( ε )] P } + (cid:88) s (cid:112) Γ s Γ s { [ ¯ f s ( ε )+ ¯ f s ( ε )] S x + 1 π [ p s ( ε ) − p s ( ε )] S y } + (cid:88) s s ¯ f s ( ε ) − Γ s ¯ f s ( ε )] S z , (C6)˙ P = (cid:88) s { s f s ( ε ) + Γ s f s ( ε )] P + 2[Γ s ¯ f s ( ε + U )+Γ s ¯ f s ( ε + U )] P − [Γ s ¯ f s ( ε ) + Γ s f s ( ε + U )+Γ s ¯ f s ( ε ) + Γ s f s ( ε + U )] P } − (cid:88) s (cid:112) Γ s Γ s ×{ [ ¯ f s ( ε ) − f s ( ε + U ) + ¯ f s ( ε ) − f s ( ε + U )] S x − π [ p s ( ε ) − p s ( ε + U ) − p s ( ε ) + p s ( ε + U )] S y }− (cid:88) s s ¯ f s ( ε ) + Γ s f s ( ε + U ) − Γ s ¯ f s ( ε ) − Γ s f s ( ε + U )] S z , (C7)˙ P = (cid:88) s { [Γ s f s ( ε + U ) + Γ s f s ( ε + U )] P − s × ¯ f s ( ε + U ) + Γ s ¯ f s ( ε + U )] P } − (cid:88) s (cid:112) Γ s Γ s ×{ [ f s ( ε + U ) + f s ( ε + U )] S x − π [ p s ( ε + U ) − p s ( ε + U )] S y } + (cid:88) s s f s ( ε + U ) − Γ s f s ( ε + U )] S z . (C8)
1. Orbital degeneracy
When considering δt (cid:54) = 0 but δε = 0, the equations forthe spin components reduce to˙ S x = − (cid:88) s (Γ s + Γ s )[ ¯ f s ( ε ) + f s ( ε + U )] S x + (cid:88) s (Γ s − Γ s ) 1 π [ p s ( ε ) + p s ( ε + U )] S y + (cid:88) s (cid:112) Γ s Γ s { f s ( ε ) × P − [ ¯ f s ( ε ) − f s ( ε + U )] P − f s ( ε + U ) P } , (C9)˙ S y = (cid:88) s (Γ s − Γ s ) 1 π [ p s ( ε ) + p s ( ε + U )] S x − (cid:88) s (Γ s +Γ s )[ ¯ f s ( ε ) + f s ( ε + U )] S y − (cid:88) s (cid:112) Γ s Γ s π [ p s ( ε )+ p s ( ε + U )] S z , (C10)1˙ S z = (cid:88) s (cid:112) Γ s Γ s π [ p s ( ε ) + p s ( ε + U )] S y − (cid:88) s (Γ s +Γ s )[ ¯ f s ( ε ) + f s ( ε + U )] S z + (cid:88) s (Γ s − Γ s ) { f s ( ε ) × P − [ ¯ f s ( ε ) − f s ( ε + U )] P − ¯ f s ( ε + U ) P } , (C11)while for the populations, the equations become˙ P = (cid:88) s { (Γ s + Γ s ) ¯ f s ( ε ) P − s + Γ s ) f s ( ε ) P } + (cid:88) s (cid:112) Γ s Γ s ¯ f s ( ε ) S x + (cid:88) s s − Γ s ) ¯ f s ( ε ) S z , (C12)˙ P = (cid:88) s { s + Γ s ) f s ( ε ) P − (Γ s + Γ s )[ ¯ f s ( ε )+ f s ( ε + U )] P + 2(Γ s + Γ s ) ¯ f s ( ε + U ) P }− (cid:88) s (cid:112) Γ s Γ s [ ¯ f s ( ε ) − f s ( ε + U )] S x − (cid:88) s s − Γ s )[ ¯ f s ( ε ) − f s ( ε + U )] S z , (C13) ˙ P = (cid:88) s { (Γ s + Γ s ) f s ( ε + U ) P − s + Γ s ) × ¯ f s ( ε + U ) P } − (cid:88) s (cid:112) Γ s Γ s f s ( ε + U ) S x + (cid:88) s s − Γ s ) f s ( ε + U ) S z . (C14)Compared with the symmetric case [see Eqs. (19)–(24)], the small asymmetry in tunnelling rates, Γ s − Γ s ∼ δt , couples { S x , P , P } to { S y , S z } . Specifically, S y and S x are coupled via principal values, p s ( ε ) and p s ( ε + U ), while S z is coupled to P , P , P via Fermi-Dirac distribution functions. This means that the tun-nelling asymmetry not only induces an additional effec-tive magnetic field B z that causes spin rotations in the xy -plane, but also allows spin accumulation in the z di-rection. C. A. Stafford, D. M. Cardamone, and S. Mazumdar,
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